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have started closed cover, for the moment mainly in order to record references.
In respect of the other thread regarding compact objects in $Sh(Mfld)$, might it be worth recording the simple result that open covers of compact, not necessarily boundaryless, manifolds are refined by (finite) closed covers such that pairwise intersections have non-empty interior? I can do it myself, in a little while, if so.
Hi David,
I assume you mean this here as a rethorical question:
might it be worth recording the simple result that […]
?
If not, then I don’t understand. What are you worried about?
Hmm, not sure. I asked on a whim.
It is a big problem in the noncommutative geometry a la Connes how to replace the condition that every point is in the interior of some element of the cover. Namely if one deals with such open covers than the usual conditions like local triviality etc. are well represented so this condition is essential (otherwise one can get lower-dimensional members of the cover and then the local triviality becomes much weaker condition and similarly for Čech cocycles). Namely in operator algebras closed sets are easy to achieve – just divide by corresponding ideals. My proposal is to work with open covers represented by pro-C-star-algebras which come as inverse limits of such closed sets, like one is exhausting an open set by increasing family of closed sets (direct limits of sets becomes an inverse limit of algebras!). But the problem with interior still stays in that approach. On the algebraic side one has at least echo of Zariski topology via flat localization theory, what is where I made (in Hopf-coaction setup) much of my career on, but it is difficult to find analogues on the operator algebra side. Woronowicz suggested to me using affiliated elements of C-star-algebra or multiplier algebra instead of pro-C-star algebra as in the Zariski examples one has unboundedness effects when going to the singularities of inverted set of observables.
@Zoran, I know it is off topic, but I presume you have seen the paper by Snigdhayan Mahanta http://arxiv.org/abs/0906.5400 (he’s a postdoc at Adelaide).
My proposal is to work with open covers represented by pro-C-star-algebras which come as inverse limits of such closed sets, like one is exhausting an open set by increasing family of closed sets
Does it work the other way around? Can you consider limits of open sets as they shrink down to a closed set? This is a fruitful way of thinking about paths in topological stacks. Naively they are represented by anafunctors which rely on open covers of $[0,1]$ for their definition, but one can really just take a closed cover as we are considering here; if one doesn’t care about smoothness, it is permissible to ’pass to the limit’ and allow overlaps to be lower-dimensional. But in general, one can take a system of shrinking open covers.
David: Surely it works other way around but the conditions on interior are likely to be easier to describe in the situation I describe: every point is in some closed set of the sequence/net, and if it is in some it is in every later/bigger one. When one goes down one can shrink size in the limit, what seems a bit more difficult to control. I know Mahanta from Max Planck days (and some correspondence earlier than that) and I knew some of his papers. It seems I missed this one, thank you.
Tim Porter: this Mahanta’s paper speculates at some point a possibility for noncommutative proper homotopy theory from an extension of his setup.
Hmm, not sure. I asked on a whim.
In any case, I have now added a remark to that effect.
I corrected Jankovič to Janković (these are different phonems).
@Zoran. Thanks. That looks very interesting as a paper and an idea. I will have to look into it in more detail and see if it is worth adding some stuff to the lab about it.
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